General Topology with the introduction to the theory of homotopy is an
essential part of mathematics, necessary to continuethe studies in several areas of mathematics,
for exampleanalysis, functional
analysis,theory of measure and
probability. The aim is to provide the basic concepts, fundamental theorems and
applications to analysis.

2.Pre-requisites in terms of knowledge, skills and social competences
(where relevant)

Basics of analysis and elementary set theory.

3.Module learning outcomes in terms of knowledge, skills and social competences
and their reference to programme learning outcomes

be able to
applyTychonoff’s theorem, fundamental
metrization theorems to constructseveral mathematical objects,Stone’a-Cech compactifications and applications for the selected
topics in (functional) analysis.This
should provide technics to recognize classes of topological spaces which are
homeomorphic.Students should be able
to use Baire’s theorem to provide several examples of topological spaces with
specific properties, they should know howto apply mentioned topological concepts to study the topology of
spaces of continuous functions C(X)overcompacta X and, more
general, over Tychonoff spaces X, with the pointwise and compact-open
topology, respectively. Theorems of Nagata, Ascoli, Stone-Weierstrass about
spaces C(X)will be presented and
proved with possible applications. Students should recognize differencesbetween metric complete spaces, completely
metrizable spaces, Polish spaces, Cech-complete spaces

KMAT2_U08,
KMAT2_U14

…E_03

be able to use and
recognize the importance ofseveral
separation axioms, as well as,theorems which describe such concepts.Students will know the Urysohn’stheoremwith possible
applications,theorems dealing with
problems about extensions of maps, like Tietze and Dugundji’s extensions
theorems. Retracts and extenders will be discussed.

KMAT2_U02,KMAT2_U18,

KMAT2_W08

E_04

be able to use
homotopic maps.

KMAT2_U18

E_05

be able to verify
the difference betweenconnected and
arcwise connected spaces.

KMAT2_U02

E_06

be able to explain
the concept of the fundamental group (or first homotopy group) of the pair
(X, x), where X is a topological space and x an element of X.

Concept of a topological space,
open sets, closed sets, interior, closure,cluster points, base, sub-base, several methods to constructtopological spaces, for example topology
given by metric spaces, examples and motivations from analysis.

Topological products,
Tychonoff’s theorem, applications. The Stone-Cech compactification, examples
including the Stone-Cech compactification of the space of naturals numbers,
applications to the theory of continuous functions C(X) over compacta X.

E_01,
E_02

TK_4

Axioms of separation, Urysohn’s
theorem with applications,Tietze and
Dugundji’s extension theorems and applications. Spaces satisfying the first
and the second axioms of countability. Urysohn’s metrization theorem,